Computed tomography techniques including X-ray Computed Tomography (CT), single photon emission computed tomography (SPECT), positron emission tomography (PET), synchrotron radiation, etc. are well-established imaging modalities. Systems using these imaging modalities (referred to as tomographic imaging systems) are used in applications such as medical, biomedical, security, and industrial, with the scanned objects being patients, small animals, material or tissue samples, pieces of luggage, or manufactured parts.
A computational process that is commonly used in tomographic imaging systems is reprojection, which, for a given two-dimension (2D) or three dimensional (3D) image, produces a set of numbers known as projections, which are closely related to the set of measurements that would be measured by a detector array in a tomographic system. Another commonly used computational process is backprojection, which is the adjoint or transpose of reprojection, and which, for a given set of projections or pre-processed projections, creates a corresponding image.
The accuracy with which the operations of reprojection and backprojection are performed affects the quality of images produced by tomographic imaging systems. Also, reprojection and backprojection are some of the most computationally demanding operations in tomographic imaging systems. For these reasons, much effort has been devoted to the development of reprojection and backprojection methods that are both accurate and computationally efficient when implemented on computers or special-purpose hardware.
The Siddon method [R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys., vol. 12, no. 2, pp. 252-255, 1985] is a widely used technique for performing the reprojection operation. The method operates by tracing the x-ray trajectory from the source to the center of the detector. As it traverses the image volume, it calculates the intersection length of the ray with a given voxel as its contribution to the detector. By tracing along the ray, only the relevant voxels that contribute to a given ray are processed. Also, certain geometric calculations are made easier by continuing along the ray. However, this method only produces the ‘pencil-ray’ projection calculations. The pencil-ray calculations can become complex to account for voxel sizes that smaller than a detector, because voxels are evenly not passed through by the ray. Ray samples must be fine enough to ensure farthest voxels are passed through. The failure to traverse voxels can create significant artifacts. The pencil-ray calculations also do not provide projections for integrating detectors, that is, detectors that integrate the projected values over the detector element area. The pencil-ray calculations therefore fail to provide an accurate model of the physical measurements in a system with integrating detectors.
The distance-driven (DD) reprojector [B. De Man and S. Basu, “Distance-driven projection and backprojection in three dimensions,” Phys. Med. Biol., vol. 49, no. 11, pp. 2463-2475, 2004] is another widely used reprojection technique. This method is attractive because it incorporates a model of the voxel's physical extent, along with integration over a detector area, and does so in a computationally efficient manner. It achieves this efficiency by effectively collapsing one of the dimensions of the image volume as it processes the data. The DD method processes the image volume in x-z or y-z slabs. However, when processing a given slab, it considers only the two dimensional x-z or y-z cross section of the voxels. The ‘slant’ of the rays is approximately compensated for by a 1/cos(α) factor (where α is the angle the ray makes with the slab), but as a result of this approximation, the method accuracy degrades as the ray's angle approaches 45 degrees.
A more recently proposed method is the Separable Footprint (SF) reprojector [Y. Long, J. Fessler, and J Balter, “3D Forward and Backprojection for X-Ray CT Using Separable Footprints,” IEEE Trans. Med. Imag., vol 29, pp 1839-1850, 2010]. It aims to overcome the inaccuracies of the DD methods by better modeling of the physical extent of the voxel basis. Unlike the DD method, the SF method is a variant of the so-called ‘splatting’ techniques, which operate in the detector domain, representing the projection of the voxel with a footprint which is scaled and stretched depending on the voxel's location in the image domain. Earlier approaches to splatting used rotationally invariant footprints to ease the computational requirements of the method. The SF approach represents the footprint in a separable way, as a function of the detector rows and channels. The SF method operates on an indirect representation of the voxel in the detector domain and not upon a direct representation in the image space that accounts for the overlap of the rays and voxels.
The various reprojectors described above also have corresponding backprojection methods, with related advantages and drawbacks.